Open problems in “Finite Geometry”: Spreads of Hermitian varieties

Here is another problem in finite geometry that I personally find intriguing, but I’m sure many others in the community find is also one of the big-ish problems in our area. It concerns spreads of Hermitian varieties, and I guess this question goes back to Segre’s 200-page manuscript “Forme e geometrie hermitiane, con particolare riguardo al caso finito” (Forms and hermitian geometries, with particular regard to the finite case).

A Hermitian variety is one of the central objects in finite and algebraic geometry. We begin with a field F having an involutory automorphism \sigma, and an Hermitian matrix A; so A^\sigma=A^T (the transpose of A). Now consider the variety defined by the following equation

X^\sigma A X^T=0,\quad X=(X_1,X_2,\ldots,X_n).

So for example, we could have F=\mathbb{C} and \sigma being complex conjugation. We could take A to be the identity matrix and we would obtain the simplest Hermitian variety. We will now consider non-degenerate Hermitian varieties, where A is invertible, and let us suppose now that F is a finite field. So now F has square order q^2 and \sigma is nothing other than the map x\mapsto x^q. It turns out that for a given dimension and order of the finite field, any pair of non-degenerate Hermitian varieties are isometric; that is, if they are defined by the matrices A_1 and A_2, then there is an invertible matrix P such that P^\sigma A_1 P^T=A_2. So from now on, we will denote the non-degenerate Hermitian variety of PG(n-1,q^2) by H(n-1,q^2).

Not only can we think of the Hermitian variety as a set of projective points of PG(n-1, q^2), but we can also look at higher-dimensional subspaces which are contained in the variety. What we mean by “contained” is that all of the projective points of the given subspace are points of the variety. It turns out that not every dimension is possible for a contained subspace. For n even, the largest such algebraic dimension is n/2, whilst for n odd, this largest dimension is (n-1)/2. This value is called the rank of the Hermitian variety. A fully contained subspace attaining this algebraic dimension will be called a maximal in what follows.

Segre asked the question whether it was possible to find a set of maximals of H(d,q^2) which form a partition of the points of H(d,q^2). This is called a spread of the Hermitian variety, and spreads arise in other contexts in projective and polar geometry.

It is tedious to calculate that the number of points of H(d,q^2) is

\frac{(q^{d+1}+(-1)^d)(q^d-(-1)^d)}{q^2-1}

and the number of points in a maximal is \frac{q^{d+1}-1}{q^2-1} for d odd and \frac{q^{d}-1}{q^2-1} for d even. So it turns out that the size of a spread is q^d+1 for d odd and q^{d+1}+1 for d even.

Segre shows in the paper mentioned above, that there is no spread of H(3,q^2), for every prime power q. The following amazing result was first proved by Jef Thas (1992).

Theorem: For d odd, the Hermitian variety H(d,q^2) has no spread.

So this leaves the case d even. The only result we have to date is a computational result by Andries Brouwer (unpublished): there is no spread of H(4,4).

Does there exist a spread of H(d,q^2) when d is even?

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